$4\mathrm{.}1$ What does the word complement mean about orthogonal subspaces?

$4\mathrm{.}2$ If $V$ is a subspace of the $7-$dimensional space $R^7,$ the dimensions of $V$ and its orthogonal complement add to

$4\mathrm{.}3$ The projection of $b$ onto the line through $a$ is the vector

$4\mathrm{.}4$ The projection matrix onto the line through $a$ is $P=$

$4\mathrm{.}5$ The key equation to project $b$ onto the column space of $A$ is the normal equation

$4\mathrm{.}6$ The matrix $A^{T}A$ is invertible when the columns of A are

$4\mathrm{.}7$ The least squares solution to $Ax=b$ minimizes what error function?